Set R contains five numbers that have an average value of 55

{\(a_1\), \(a_2\), \(a_3\), \(a_4\), \(a_5\)}
As mean of 5 numbers is 55 then the sum of these numbers is \(5*55=275\);
The median of the set is equal to the mean --> \(mean=median=a_3=55\);
The largest number in the set is equal to 20 more than three times the smallest number --> \(a_5=3a_1+20\).

So our set is {\(a_1\), \(a_2\), \(55\), \(a_4\), \(3a_1+20\)} and \(a_1+a_2+55+a_4+3a_1+20=275\).

The range of a set is the difference between the largest and smallest elements of a set.

\(Range=a_5-a_1=3a_1+20-a_1=2a_1+20\) --> so to maximize the range we should maximize the value of \(a_1\) and to maximize \(a_1\) we should minimize all other terms so \(a_2\) and \(a_4\).

Min possible value of \(a_2\) is \(a_1\) and min possible value of \(a_4\) is \(median=a_3=55\) --> set becomes: {\(a_1\), \(a_1\), \(55\), \(55\), \(3a_1+20\)} --> \(a_1+a_1+55+55+3a_1+20=275\) --> \(a_1=29\) --> \(Range=2a_1+20=78\)

Answer: A.